Best Known (61, 61+15, s)-Nets in Base 2
(61, 61+15, 180)-Net over F2 — Constructive and digital
Digital (61, 76, 180)-net over F2, using
- trace code for nets [i] based on digital (4, 19, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
(61, 61+15, 363)-Net over F2 — Digital
Digital (61, 76, 363)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(276, 363, F2, 2, 15) (dual of [(363, 2), 650, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(276, 524, F2, 2, 15) (dual of [(524, 2), 972, 16]-NRT-code), using
- 21 times duplication [i] based on linear OOA(275, 524, F2, 2, 15) (dual of [(524, 2), 973, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(275, 1048, F2, 15) (dual of [1048, 973, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(274, 1047, F2, 15) (dual of [1047, 973, 16]-code), using
- adding a parity check bit [i] based on linear OA(273, 1046, F2, 14) (dual of [1046, 973, 15]-code), using
- construction XX applied to C1 = C([1021,10]), C2 = C([1,12]), C3 = C1 + C2 = C([1,10]), and C∩ = C1 ∩ C2 = C([1021,12]) [i] based on
- linear OA(261, 1023, F2, 13) (dual of [1023, 962, 14]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(260, 1023, F2, 12) (dual of [1023, 963, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(271, 1023, F2, 15) (dual of [1023, 952, 16]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,12}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(250, 1023, F2, 10) (dual of [1023, 973, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(21, 11, F2, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to C1 = C([1021,10]), C2 = C([1,12]), C3 = C1 + C2 = C([1,10]), and C∩ = C1 ∩ C2 = C([1021,12]) [i] based on
- adding a parity check bit [i] based on linear OA(273, 1046, F2, 14) (dual of [1046, 973, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(274, 1047, F2, 15) (dual of [1047, 973, 16]-code), using
- OOA 2-folding [i] based on linear OA(275, 1048, F2, 15) (dual of [1048, 973, 16]-code), using
- 21 times duplication [i] based on linear OOA(275, 524, F2, 2, 15) (dual of [(524, 2), 973, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(276, 524, F2, 2, 15) (dual of [(524, 2), 972, 16]-NRT-code), using
(61, 61+15, 5668)-Net in Base 2 — Upper bound on s
There is no (61, 76, 5669)-net in base 2, because
- 1 times m-reduction [i] would yield (61, 75, 5669)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 37797 963409 496602 792920 > 275 [i]